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2011 | 9 | 6 | 1349-1353
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One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

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Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A v; v) is 1. We then give a criterion for R to be regular.
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Bibliografia
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  • [3] Debremaeker R., Van Lierde V., On adjacent ideals in two-dimensional rational singularities, Comm. Algebra, 2010, 38(1), 308–331 http://dx.doi.org/10.1080/00927870903392476
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  • [12] Van Lierde V., A mixed multiplicity formula for complete ideals in 2-dimensional rational singularities Proc. Amer. Math. Soc., 2010, 138(12), 4197–4204 http://dx.doi.org/10.1090/S0002-9939-2010-10455-0
  • [13] Van Lierde V., Degree functions and projectively full ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal, J. Pure Appl. Algebra, 2010, 214(5), 512–518 http://dx.doi.org/10.1016/j.jpaa.2009.06.002
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  • [15] Zariski O., Samuel P., Commutative Algebra. II, The University Series in Higher Mathematics, Van Nostrand, Princeton-Toronto-London-New York, 1960
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-011-0095-y
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